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Note: While this question as phrased in the title is somewhat subjective, what I'm looking for as an answer should be specific enough to still have a clear/valid answer(s) to it. Also, apologies in advance if this is somewhat long winded but I feel the full context of my question is needed for clarity.

First let me provide some context. When it comes to learning new areas of math (or in this case, arguably, more fundamental/abstract areas), one problem all students face is 'vision'. Taking new concepts and adding them to old ones is often difficult since to do that requires seeing how things interrelate. However your typical education in math usually leaves you vastly unprepared to do this.

In a sense, most modern math education teaches students backwards. Most of the maths that students learn is gutted of the more important (and abstract) details. Details such as the 'real' proofs of certain theorems being replaced with a shorthand version that, while proves the part of the concept being focused on, ignores the parts that give it its foundation. "Proofs in a vacuum" as I call them since they usually don't bother proving (sometimes not even mentioning) other concepts that the main idea depends on.

This leads me to my problem. I have a rather strong foundation in multiple areas of math (particularly those often relevant to physics, such as calculus), but since my education has focused on results/applications rather then math as a whole I'm left without the 'glue' that ties it all together.

The language of set theory (as well as basic forms of logic) is still new to me. While I have a strong (enough) foundation in the basics after recently completing a class in discrete math, taking the concepts (and building onto them where needed) and applying them so as to add to my understanding of calculus for example is still difficult.

My question is this: What are some methods/pet projects I should try an implement to 'break out' of this novice phase as far set theory is concerned? To bridge the gap between knowing basic ideas concerning sets and relations between them and concepts from calculus for example? I don't want to look at an equation and see an equation. I wanna be able to see it as a more abstract entity.

Keep in mind what I'm looking for in an answer is:

  • Specific topics to research that show how sets and relations are used to construct other others of math
  • Go to examples for me to play around with so as to see how set theory is applied in practice (for example, how relations can be used to construct basic operations like addition, or differential operators).
  • Examples of a common (but not obvious) mistakes people tend to make applying set theory to describe common concepts in other area of math.
shardulc
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Suisami1990
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    It's better to teach mathematics gradually in my opinion. Overloading students with extreme abstraction before they even know how to integrate is probably counter-productive. I find it better to introduce them to concepts that they can understand easily and then introduce the "reasoning" behind the ideas afterwards. You wouldn't teach High School students the analysis behind integrable functions before teaching them how to evaluate $\int x dx$ – Edward Evans Dec 25 '16 at 23:42
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    See P Halmos: Naive set theory. – kjetil b halvorsen Dec 25 '16 at 23:44
  • I agree with @kjetilbhalvorsen. Technically, you can embed "all of math" in to set theory. But It should be noted that depending on your area of expertise, you may not need anything more than the details in Halmos. Also, once you have read that and mastered the material to a reasonable degree, you may want to take look at how the real numbers are constructed and the long line (an example that pops up in topology.) –  Dec 27 '16 at 02:57
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    Perhaps a text on general topology as well as one on set theory. Topology often focuses on certain families of sets, families of functions,etc. and forms part of the fundamentals behind much analysis, for example the Heine-Borel theorem, the theory of Lebesgue integration, etc. – DanielWainfleet Dec 29 '16 at 10:39
  • I second the recommendation of @user254665 on general topology, with a specific recommendation of Munkres' book "Topology", which has a detailed introductory section on naive set theory that he then goes on to apply throughout the book. – Lee Mosher Dec 31 '16 at 17:33
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    As a logician and occasional set theorist, I'd caution against viewing set theory as the "glue" that holds mathematics together. Math does fine on its own, for the most part; set theory can have a philosophical role in saying what mathematical objects "are", but this has been roundly criticized (and I agree with the criticism). Its main role foundationally is to provide a common set of axioms for mathematical practice; see here for more on this theme. I think set theory is best motivated non-foundationally: – Noah Schweber Dec 31 '16 at 17:39
  • either through its own interest, or through constructions in mathematics that invoke it in a serious way. Based on your question, I suggest the latter. One natural point for this is topology; besides being naturally a subject about sets, many counterexamples in topology are very set-theoretic in nature. Another point of attack is nonstandard analysis, which is developed via ultrapowers, and uses the axiom of choice subtly. Nonstandard analysis has been useful in functional analysis, and in providing an alternative approach to calculus. – Noah Schweber Dec 31 '16 at 17:41

2 Answers2

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If I understand your question correctly, although you have embedded a critique of how abstract math is taught, your question is really about how you can shore up your own understanding of abstract math via an enhanced understanding of set theory.

I would suggest that you learn about how the real number system is built up in steps from Peano's axioms for the natural numbers. There are three steps:

  1. From Peano's axioms for the natural numbers to the integers.
  2. From the integers to the rational numbers.
  3. From the rational numbers to the real numbers.

There is one still deeper step:

  1. From the ZFC axioms of set theory to Peano's axioms for the natural numbers (via the construction of the ordinal numbers).

I cannot recommend a single source to you for all four of these steps, although steps 2, 3, 4 at least are in many books at the advanced undergraduate or early graduate level, with steps 2 and 3 more likely to be in number theory books and steps 3 and 4 more likely to be in advanced calculus books. For Step 2, I like the closing two chapters of the book "Theory of Numbers" by B. M. Stewart. For Step 1, I like the early sections of Cohen's slim little atom bomb of a book "Set theory and the continuum hypothesis".

Lee Mosher
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It sounds like you want to establish a better foundational understanding of mathematics in order to help you with your applied work in areas such as calculus and physics, and perhaps for personal satisfaction. Set theory is indeed often considered the "glue" or underpinning for comprehending the whole of math, and a careful study of it will assuredly help you develop a more interconnected view of many different areas of research. So I would simply suggest the following. Get a basic book on set theory such as Charles Pinter's "A Book of Set Theory" and study it carefully. It's not that difficult. Of all the set theory books I've seen, this one is exceptional in terms of clarity and coverage and could provide you with a better fundamental understanding of mathematics as a whole. Set theory also segues nicely into other areas of research such topology and real analysis, which are perhaps most relevant for studying the foundations of calculus. Two additional options to consider are Paul R. Halmos' "Naive Set Theory", and Hamilton and Landin's "Set Theory: The Structure of Arithemetic".

Mike
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